
The Condat–Vũ algorithm [1]}, [2]} exists in a primal
and a dual variant.
We discuss extensions of the two algorithms
that use Bregman proximal operators in the primal and dual updates.
The Bregman primal Condat–Vũ algorithm first appeared
in [3]}, and is also a special case
of the algorithm proposed in [4]}
for a more general convex–concave saddle point problem.
We give a new derivation of this method and its dual variant,
by applying the Bregman proximal point method
to the primal–dual optimality conditions.
Based on the interpretation, we provide a unified framework
for the convergence analysis of the two variants,
and show an \(O(1/k)\)  ergodic convergence rate,
which is consistent with previous results for Euclidean proximal operators
in [1]}, [2]} and Bregman proximal operators in [3]}.
We also give a convergence result for the primal and dual iterates.

We propose an easily implemented backtracking line search technique
for selecting stepsizes in the Bregman dual Condat–Vũ
algorithm for problems with equality constraints.
The proposed backtracking procedure is similar to the technique
in [8]} for the special setting of PDHG with Euclidean proximal
operators, but has some important differences even in this special case.
We give a detailed analysis of the algorithm
with line search and recover the \(O(1/k)\)  ergodic rate of convergence
for related algorithms in [8]}, [10]}.

We propose a Bregman extension for PD3O
and establish an ergodic convergence result.

